相关论文: Why prove things?

Interactive proof assistants make it possible for ordinary mathematicians to write definitions and theorems in a formal proof language, like a programming language, so that a computer can parse them and check them against the rules of a…

We discuss a practical method for assessing mathematical proof online. We examine the use of faded worked examples and reading comprehension questions to understand proof. By breaking down a given proof, we formulate a checklist that can be…

We develop a semantics for logics of imperfect information with respect to general models. Then we build a proof system and prove its soundness and completeness with respect to this semantics.

In many everyday categories (sets, spaces, modules, ...) objects can be both added and multiplied. The arithmetic of such objects is a challenge because there is usually no subtraction. We prove a family of cases of the following principle:…

A theory graph is a network of axiomatic theories connected with meaning-preserving mappings called theory morphisms. Theory graphs are well suited for organizing large bodies of mathematical knowledge. Traditional and formal proofs do not…

Mathematics can serve many functions in physics. It can provide a computational system, reflect a physical idea, conveniently encode a rule, and so forth. A physics student thus has many different options for using mathematics in his…

The generally accepted wisdom in computational circles is that pure proof verification is a solved problem and that the computationally hard elements and fertile areas of study lie in proof discovery. This wisdom presumably does hold for…

This is an explanation and defense of "mathematical conceptualism" for a general mathematical and philosophical audience. I make a case that it is cogent, rigorous, attractive, and better suited to ordinary mathematical practice than all…

It has been shown that a functional interpretation of proofs in mathematical analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the computation of optimal strategies in…

While model checking has often been considered as a practical alternative to building formal proofs, we argue here that the theory of sequent calculus proofs can be used to provide an appealing foundation for model checking. Since the…

The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…

Engineering needs mathematics, but the converse is also increasingly evident. Indeed, mathematics is still recovering from the drawbacks of several "reforms". Encouraging is the revived interest in proofs indicated by various recent…

The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which…

The capability of making explainable inferences regarding physical processes has long been desired. One fundamental physical process is object motion. Inferring what causes the motion of a group of objects can even be a challenging task for…

We present a prototype of an integrated reasoning environment for educational purposes. The presented tool is a fragment of a proof assistant and automated theorem prover. We describe the existing and planned functionality of the theorem…

When a proposition has no proof in an inference system, it is sometimes useful to build a counter-proof explaining, step by step, the reason of this non-provability. In general, this counter-proof is a (possibly) infinite co-inductive proof…

We define the concept of collaborative theorem proving and outline our plan to make it a reality. We believe that a successful implementation of collaborative theorem proving is a necessary prerequisite for the formal verification of large…

We lay the groundwork for a formal framework that studies scientific theories and can serve as a unified foundation for the different theories within physics. We define a scientific theory as a set of verifiable statements, assertions that…

Evidential reasoning is cast as the problem of simplifying the evidence-hypothesis relation and constructing combination formulas that possess certain testable properties. Important classes of evidence as identifiers, annihilators, and…

In this paper, we consider the problem of learning a first-order theorem prover that uses a representation of beliefs in mathematical claims to construct proofs. The inspiration for doing so comes from the practices of human mathematicians…